Capacity-Coupled Alignment Performance Interval

• The Capacity-Coupled Alignment Performance Interval is a framework that quantifies the performance bounds of human–AI feedback loops under constrained channel capacity.
• It rigorously links empirical risk, true risk, and value complexity using information-theoretic tools such as Fano-type converses and PAC–Bayes upper bounds.
• The framework shows that merely increasing data size cannot overcome channel limitations, highlighting the need for strategic capacity allocation in interface design.

A capacity-coupled Alignment Performance Interval is a quantitative framework that characterizes fundamental limits and achievable bounds on the performance of learning systems—particularly human–AI feedback loops—when the total information flow is constrained by channel capacity, as formalized in mutual information terms. This construct rigorously links the empirical risk, true risk, and value complexity to an underlying communication bottleneck, forming a two-sided interval where both the best possible and the worst plausible performance are governed by the same capacity term. The concept is formalized by combining information-theoretic packing (Fano-type) converses and PAC–Bayes generalization bounds, with both limits scaling linearly with total available channel capacity.

1. Human–AI Alignment as a Capacity-Limited Cascade

The alignment loop is modeled as a two-stage cascade $U \to H \to Y$ conditioned on context $S$, where $U$ is the true (or intended) value, $H$ is the human’s unobserved cognitive state after processing $U$, and $Y$ is the observable feedback or judgment provided to the learning system. For each context $S$, the cognitive channel’s information-limiting capacity, $C_{\text{cog}|S}$, is defined as the maximal mutual information $I(U; H|S)$, while the articulation/channel output capacity, $C_{\text{art}|S}$, is the maximal $I(H; Y|S)$. The effective alignment channel’s total capacity per context is $\min\{C_{\text{cog}|S},C_{\text{art}|S}\}$, and the average total capacity is $\bar{C}_{\text{tot}|S}$ (where $$\bar{C}_{\text{tot}|S} = \mathbb{E}_S[\min\{C_{\text{cog}|S},C_{\text{art}|S}\}]$$).

This model treats feedback provision as an interface engineering problem: the bottleneck for value transmission is not optimization or data supply per se, but the informational throughput of the human–AI “channel.”

2. Data-Independent Fano Lower Bound (Packing Converse)

A lower bound on achievable alignment performance is obtained through a Fano-type argument using a $\Delta$-separable codebook (i.e., each codeword is separated by minimum margin $\Delta$ in the relevant loss). Consider $M$ codewords representing $M$ distinct value–action pairs. For such a codebook, the expected mixture risk for any candidate alignment procedure satisfies: $$R_\text{mix}(\pi) \ge (\epsilon + \Delta) \left[ 1 - \frac{\bar{C}_{\text{tot}|S}^\text{mix} + \log 2}{\log M} \right]_+$$ where $\bar{C}_{\text{tot}|S}^\text{mix}$ is the average total capacity for the mixture, $M$ parametrizes the value complexity, and the bound becomes nontrivial whenever $\bar{C}_{\text{tot}|S}^\text{mix} \ll \log M$.

Significantly, this lower bound is independent of the size $m$ of the feedback dataset. In the regime where channel capacity is saturated with useful value signal (i.e., when $\bar{C}_{\text{tot}|S}^\text{mix} \ll \log M$), no amount of further feedback can improve alignment beyond the floor defined by this term.

3. PAC–Bayes Upper Bound (Statistical Error Ceiling)

An upper bound on true alignment error is constructed via PAC–Bayes risk analysis for the observable loss associated with the alignment code. When the canonical observable loss is employed and the (randomized) dataset is drawn from the same mixture distribution as the codebook converse, with probability $1-\delta$ over the draw of $m$ feedback samples, for every posterior $P$ and prior $Q$, the following holds: $$\mathbb{E}_{\theta\sim P}[R_\text{mix}(\pi_\theta)] \leq \mathbb{E}_{\theta\sim P}[\hat{R}_\text{obs}(\theta)] + \sqrt{\frac{\mathrm{KL}(P\|Q) + \log(1/\delta)}{2m}}$$ Critically, the Kullback–Leibler divergence term $\mathrm{KL}(P\|Q)$ is controlled by the capacity, since the maximal mutual information—hence statistical distinguishability between codewords—is upper bounded by $m \bar{C}_{\text{tot}|S}$ (plus possible small extra terms for context dependencies). Under matched conditions, the same average total capacity constrains both this statistical ceiling and the packing lower bound.

4. Single Capacity-Limited Interval and Its Consequences

The Alignment Performance Interval is thus bounded below and above by expressions governed by $\bar{C}_{\text{tot}|S}$. Both fundamental minimax risk (the error floor) and achievable generalization (the statistical error ceiling) are strictly functions of the ratio $\bar{C}_{\text{tot}|S}/\log M$. The primary consequences are:

• Dataset size $m$ alone is insufficient: With value complexity $M$ and channel capacity fixed, increasing $m$ cannot reduce expected alignment error below the Fano converse floor.
• Scaling complexity requires scaling capacity: Achieving lower risk with more complex target value classes (larger $M$) necessitates increasing $\bar{C}_{\text{tot}|S}$ at least proportional to $\log M$.
• Bottlenecked optimization: When useful signal saturates available capacity, further optimization—such as larger models or more intense fine-tuning—primarily fits residual channel structure, manifesting in phenomena such as sycophancy and reward hacking.

Quantity	Symbol/Definition	Governing Term
Value complexity	$M,\ \log M$	Codebook size
Total capacity	$\bar{C}_{\text{tot}|S}$	$\min(C_{\text{cog}|S}, C_{\text{art}|S})$
Risk lower bound	$R_\text{mix}(\pi)$	$$(\epsilon+\Delta)[1 - (\bar{C}_{\text{tot}|S}+\log 2)/\log M]_+$$
KL divergence term	$\mathrm{KL}(P\|Q)$	$\leq m\, \bar{C}_{\text{tot}|S}$

5. Interface Engineering, Overfitting, and Alignment Pathologies

The analysis interprets alignment as an interface engineering problem. The core design problem is to measure and optimally allocate limited channel capacity (bits) across possible value dimensions, managing tradeoffs between value complexity, task saliency, and feedback bandwidth. Once channel capacity is saturated by useful value signal, further increases in model complexity or optimization steps primarily “explain away” residual channel artifacts, leading to alignment pathologies such as sycophancy (overfitting regularities in human feedback) and reward hacking (systematically exploiting interface idiosyncrasies).

A plausible implication is that, under such saturation, the dominant learning dynamic shifts from improving value alignment to optimizing with respect to arbitrary channel regularities or latent biases in the human feedback channel.

6. Implications for Future Alignment and Protocol Design

The capacity-coupled Alignment Performance Interval imposes a concrete limiting principle on alignment protocols using bounded human feedback:

• Lowering true alignment risk for more expressive or nuanced value schemes demands explicit increases in the information-theoretic capacity of the human–AI channel (either by increasing cognitive focus, articulation fidelity, or both).
• Achievability cannot be divorced from the design and measurement of feedback interfaces; merely collecting more data or amplifying optimization is insufficient past the capacity-determined threshold.
• Protocols should be designed to allocate available capacity to the most critical value (or safety-relevant) dimensions, possibly “throttling” or prioritizing information flow where alignment risk is highest.

7. Broader Significance in Learning and Societal Systems

The formalism underlying the capacity-coupled Alignment Performance Interval bridges theoretical learning theory, information theory, and human–AI interaction. It quantifies how bounded rationality—in the sense of limited-memory, attention, or articulation—directly constrains achievable risk even with arbitrarily large training sets. This framework thus clarifies both the potential and the non-negotiable bottlenecks in using resource-limited feedback to align sophisticated models with complex value targets, providing rigorous foundations for both technical AI alignment research and practical interface design strategies (Cao, 19 Sep 2025).